How are you looking for them to integrate the shipping costs for you?

Could you elaborate a little more on what you're looking for? I'm sure someone here could offer some good advice or point you in the right direction, but its a little unclear as to what you need here. By integrating shipping costs, do you mean including them with your purchase orders?

Here is how I would do it. First, ask where their warehouse is. Use that as your ship from address. Sign up for UPS/FED Ex/UPS, etc (who is their preferred shipper?) and do real time shipping quotes. Most likely it will be higher than what they charge based on their volume.

Here is the other issue. Except for large items, most places are going to do a flat $5 shipping or free shipping. I have a feeling you will not have enough margin from a "dropship" company like that to offer it.